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Course, academic year 2022/2023
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Projective geometry I - NMTD205
Title: Projektivní geometrie I
Guaranteed by: Department of Mathematics Education (32-KDM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: Mgr. Lukáš Krump, Ph.D.
Incompatibility : NMUG106
Interchangeability : NMUG106
Annotation -
Last update: RNDr. Jakub Staněk, Ph.D. (10.07.2020)
Construction of projective plane and projective extension of Euclidean plane. Description of conics and construction of conics from given elements.
Literature -
Last update: RNDr. Jakub Staněk, Ph.D. (10.07.2020)

Richter-Gebert, J.: Perspectives on projective geometry: a guided tour through real and complex geometry, Springer 2011

Hlavatý, V., Projektivní geometrie I. Praha, Melantrich, 1944.

Havlíček, K.: Úvod do projektivní geometrie kuželoseček. Praha, SNTL, 1956.

Syllabus -
Last update: RNDr. Jakub Staněk, Ph.D. (10.07.2020)

Projective line and plane, geometric point, homogeneous coordinates, projective extension of the affine line, affine plane, proper and improper points. Cross ratio, harmonic quadruple. Projectivity on the line, in the plane. The duality principle.

Projectivity and perspectivity of linear systems. Constructions of projectivities, perspectivities, direction line, direction point, Pappos theorem. Fixed points of a projectivity on a line. Involution. Complete quadripoint, quadrilateral.

Projective construction of conics. Construction of a tangent line, of tangent points. Construction of a projetivity on a conic. Involution on a conic.

Affine classification of regular conics, special constructions for hyperbola, parabola, ellipse. Perpendicularity, circle, constructions with an auxiliary circle.

Pascal and Brianchon theorems.

Pole and polar, conjugated poles and polars. Conjugated diameteres, foci.

 
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